If ₀^2 x(2 - x)^b ,dx = 32 7 , where b N then b =

Step-by-step solution

Let I = ₀^2 x(2 - x)^b , dx . Using the property ₀^a f(x) , dx = ₀^a f(a - x) , dx , we get: I = ₀^2 (2 - x)(2 - (2 - x))^b , dx I = ₀^2 (2 - x)x^b , dx I = ₀^2 (2x^b - x^ b+1 ) , dx Integrating the terms, we get: I = [ 2x^ b+1 b+1 - x^ b+2 b+2 ]₀^2 I = 2 2^ b+1 b+1 - 2^ b+2 b+2 I = 2^ b+2 ( 1 b+1 - 1 b+2 ) I = 2^ b+2 (b+1)(b+2) It is given that I = 32 7 . 2^ b+2 (b+1)(b+2) = 32 7 Substituting b = 6 from the options: 2⁶⁺² (6+1)(6+2) = 2^8 7 8 = 256 56 = 32 7 This satisfies the given equation. Therefore, b = 6 . Ans